Let f(x) be a function such that f(x + y) = f(x) + f(y) + xy(x + y) and lim as h approaches 0 of f(h)/h = 1. Find f'(x).

A. x^2 + 1 — Using the definition of the derivative, f'(x) = lim h to 0 of (f(x+h) - f(x))/h. Substituting the functional equation, we get f'(x) = lim h to 0 of (f(h) + xh(x+h))/h = 1 + x^2.

JEE Main · MathematicsHard

$Let f (x + y) = f (x) + f (y) + x y (x + y) and lim_{h \to 0} \frac{f ( h )}{h} = 1. Find f^{'} (x) .$

A. $x^{2} + 1$
B. $x + 1$
C. $3 x^{2} + 1$
D. $$2x + 1$$

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Correct answer: A — $x^{2} + 1$

Solution

f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h} = h \to 0 lim \frac{f ( h ) + x h ( x + h )}{h} = h \to 0 lim (\frac{f ( h )}{h} + x (x + h)) = 1 + x^{2} .

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Let f(x+y)=f(x)+f(y)+xy(x+y) and limh→0​hf(h)​=1. Find f′(x).

Solution

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More Limits & Continuity practice questions

$Let f (x + y) = f (x) + f (y) + x y (x + y) and lim_{h \to 0} \frac{f ( h )}{h} = 1. Find f^{'} (x) .$